I knew in advance that I was going to drop most of these shows, but most of them are at least worth a look. From certain perspectives, a lot of these shows are actually quite good.
The best show I stopped watching is Servant x Service which has some real appeal, but I dropped it because I didn't like Lucy. Most of the other characters are fine, though. It sort of has the same vibe as WORKING!! which I disliked. Servant x Service doesn't rub me the wrong way like WORKING!! did, but I didn't see a point in watching more Servant x Service when there are so many better shows to watch this season.
Kiniro Mosaic (Golden Mosaic) is cute. Aggressively cute. Obnoxiously cute. Well, yeah; it's a "cute girls doing cute things" show. (As mentioned previously, I'm not a big fan of those.) That's really all the show seems to have going for it. Still, I was much more impressed with the episodes I did watch than I was with the manga.
Kitakubu Katsudou Kiroku (Chronicles of the Going Home Club) is another "cute girls doing cute things" show. KKK appealed to me even less than Kiniro Mosaic because most of its humor seemed to be based on super-overblown reactions to medium-outrageous antics.
Gen'ei o Kakeru Taiyou: Il Sole Penetra le Illusioni (Day Break Illusion) isn't exactly good, but it's also not as bad as its most dismissive critics claim. Yes, it does seem designed to cash in on the popularity of Mahou Shoujo Madoka Magica, but that doesn't necessarily make it a bad show. That said, I did not find it particularly interesting, and was only watching to see if anyone got her head bitten off.
Chou Jigen Game Neptume The Animation (Hyperdimension Neptunia: The Animation) also did not strike me as a necessarily bad show, but I sensed I wouldn't be able to appreciate it without being familiar with the game first. Indeed, its most enthusiastic fans appear to be veterans of the video game. I guess that means it's a good adaptation.
If you watched the first High School DxD then you know what to expect from High School DxD NEW. It's pretty much the same deal again, but that just wasn't enough this season.
I gave Futari wa Milky Holmes (We are Milky Holmes) a couple episodes to turn itself into first-season Tantei Opera Milky Holmes, but it didn't happen so I dropped it.
Now we're in the dregs. Despite its huge popularity and strong start, Shingeki no Kyojin (Attack on Titan) really took a dive summer 2013. It doesn't help that it started off with a clip show (episode "13.5"). Notably, it was the worst clip show I have ever seen. I'm not opposed to recaps as a matter of principle. Just as I never skip OPs and EDs, I also never skip clip shows. However, the Shingeki no Kyojin recap episode was so lackluster that I have to conclude nobody responsible for it had the slightest interest in it. Then the week after, episode 14 proved so disengaging that I finally gave up on the series entirely. (This was also partially inspired by my declining opinion of the ongoing manga. I've nothing to look forward to except [SPOILERS] with a bow.) I'm forced to conclude the two episodes of Attack on Titan that I watched summer 2013 were WORSE THAN COSPRAYERS. Congratulations.
All of the shows I just mentioned are watchable on some level and arguably have positive aspects going for them. However, the summer 2013 season is too good on a whole to waste time with shows that are so flawed or otherwise uninspiring.